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Minimal k-Connected Graphs with Minimal Number of Vertices of Degree k

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A graph is k-connected if it has at least k+1 vertices and remains connected after deleting any k−1 vertices. A k-connected graph is said to be minimal if any its subgraph obtained by deleting any edge is not k-connected. W. Mader proved that any minimal k-connected graph with n vertices has at least \( \frac{\left(k-1\right)n+2k}{2k-1} \) vertices of degree k. The main result of the present paper is that any minimal k-connected graph with minimal number of vertices of degree k is isomorphic to a graph G k,T , where T is a tree the maximal vertex degree of which is at most k + 1. The graph G k,T is constructed from k disjoint copies of the tree T in the following way. If a is a vertex of T of degree j and a 1 , . . . , a k are the corresponding vertices of the copies of T, then k + 1 − j new vertices of degree k, which are adjacent to {a 1 , . . . , a k }, are added. Bibliography: 10 titles.

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Authors and Affiliations

  1. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg State University, St. Petersburg, Russia

    D. V. Karpov

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  1. D. V. Karpov
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Correspondence to D. V. Karpov.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 427, 2014, pp. 41–65.

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Karpov, D.V. Minimal k-Connected Graphs with Minimal Number of Vertices of Degree k . J Math Sci 212, 666–682 (2016). https://doi.org/10.1007/s10958-016-2697-1

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  • DOI: https://doi.org/10.1007/s10958-016-2697-1

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